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1answer
221 views

Reformulation of Grothendieck vanishing theorem

Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$? ...
1
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1answer
115 views

Sheaf cohomology of a complement of finitely many points

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $p \subset X$ a closed point in $X$. How do I compute $H^1(\mathcal{O}_{X\backslash p})$? Any reference/idea will be most welcome.
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0answers
63 views

Thom-type isomorphism on sheaf cohomology

Let $X$ be a smooth, projective surface and $T$ a finite set of points in $X$ i.e., of codimension $2$ in $X$. Is it true that $H^i(\mathcal{O}_X)=H^i(\mathcal{O}_{X\backslash T})$ for $i \ge 1$?
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1answer
140 views

Dual of a stable locally free subsheaf is a locally free quotient sheaf

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$. By [1] definition 1.2: A line bundle $L$ ...
5
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1answer
214 views

Naive question on local cohomology

Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims: $$...
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0answers
197 views

Is chapter 5 of Grothendieck (1955) related to Sheaf Cohomology?

I'm curious if the fifth and last chapter of Grothendieck's 1955/1958 University of Kansas research report General theory of fibre spaces with structure sheaf (Grothendieck Circle pdf copy; link to ...
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0answers
101 views

What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration $$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
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0answers
45 views

Continuous maps vs filtrations construction of the Leray spectral sequence

The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...
0
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1answer
89 views

Sheaf cohomology relative to a closed subspace

Let $i : A \hookrightarrow X$ be a closed subspace of a topological space $X$, and $j : Z := X \setminus A \hookrightarrow X$ denote its open complement. Given a sheaf $F$ of abelian groups on $X$, ...
1
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1answer
123 views

Commutativity between functors on sheaves of abelian groups

I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
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0answers
151 views

How to calculate : $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) $?

I try to calculate the rational cohomology algebra $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{+ \infty} \mathrm{Hdg}^{ 2 k } (...
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0answers
62 views

Convolution of sheaves on R

I am trying to understand a basic computation of convolution. Throughout, $R$ is the real line as a topological group and $k$ is some base field. I would like to understand the computation of the ...
2
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1answer
227 views

sheaf cohomology and deRham cohomology of a stack

I am reading https://arxiv.org/pdf/math/0605694.pdf to understand about (hyper)cohomology groups of stack $\mathcal{X}$ with valued in a complex of abelian sheaves $\mathcal{M}$. Let $\mathcal{F}$ be ...
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0answers
50 views

maps between two Leray spectral sequences based on maps on cochain complexes

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
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0answers
92 views

Maps between Leray spectral sequences

Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...
2
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1answer
107 views

Leray spectral sequence for continuous functions on pairs of topological spaces

Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$. The Leray spectral sequence (with complex ...
4
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1answer
210 views

On the Leray spectral sequence and sheaf cohomology

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...
3
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0answers
56 views

exact sequence of fundamental groups associated to “almost” smooth families of curves

Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
2
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0answers
122 views

An algorithm to compute coherent sheaf cohomology in projective space over a ring [closed]

EDIT: As the article was put on hold, because it was unclear what I am asking, here I put again my two questions: 1) Is the argument I used to derive the algorithm valid? The second question is a ...
3
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0answers
54 views

Canonical map in the direct image of $\mathscr{D}_X$

Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...
5
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1answer
212 views

Sheaf cohomology with support vanishes

I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset ...
1
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1answer
509 views

Inception of modern view of Sheaf Cohomology in Mathematical Literature

From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
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0answers
57 views

English translation of Faisceaux algebriques coherents J-P.S 1955 [duplicate]

Grothendieck (1957; 3. Cohomology with coefficients in a sheaf) mentions this paper [Faisceaux algebriques coherents] of J-P.Serre as fundamental. I'd really appreciate if anyone could point me to an ...
0
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1answer
128 views

The cohomology of meromorphic functions

Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...
2
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0answers
120 views

Reference for the Koszul--Malgrange Theorem

The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
6
votes
1answer
166 views

Cohomology of sheaf of Schwartz distributions with support in a submanifold

Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely ...
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0answers
162 views

Cohomology of a structure sheaf of a normal affine variety

I can't find the reference for the following fact: Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
1
vote
1answer
202 views

Cartier Divisor generated by Global Sections

Let $X$ be an integer curve of (arithmetic) genus $g=0$. (the arithmetic genus $g$ is defined by $g:= 1 -\chi_k(\mathcal{O}_X)$ where $\mathcal{O}_X$ is the structure sheaf of $X$ and $\chi_k(\mathcal{...
2
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2answers
105 views

Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?

Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$. Let $k$ be a ring and for every $...
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0answers
80 views

Question on de Rham complex with distributional coefficients

Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
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1answer
412 views

Is there a complex which computes Cech cohomology?

Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open ...
2
votes
1answer
273 views

Is any element in $H^2_{et}(X,\mathcal{O}_X^*)$ locally trivial in the Zariski topology?

Let $X$ be an algebraic variety over a field $k$ and we consider the cohomological Brauer group $H^2_{et}(X,\mathcal{O}_X^*)$. For any element $\alpha \in H^2_{et}(X,\mathcal{O}_X^*)$ and any closed ...
7
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1answer
301 views

What are the uses of coefficient systems for arithmetic cohomology theories?

In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local ...
2
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0answers
82 views

Theta divisor on compactified jacobian of nodal curve

Let $X$ be a Nodal curve. Let $\bar{J}(X)$ be compactified Jacobian (rank one torsion free sheaf of degree one) and $\Theta$ denote the theta divisor in $J$. How to compute $H^0(\bar{J}(X);\Theta^k)$, ...
6
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0answers
146 views

Defining the Euler class in different ways

Let $\pi: E\to M$ be a rank two real vector bundle over a manifold $M$. Bott and Tu defines the Euler class by: giving $M$ a Riemannian structure, taking a trivializing chart $U_\alpha$ of $M$, ...
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0answers
101 views

Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
8
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0answers
276 views

Leray-Hirsch theorem for Dolbeault cohomology

In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this: Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...
8
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1answer
503 views

Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?

Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$. Do the cohomology groups $H^i(Y,\mathcal{O}...
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0answers
70 views

Local cohomology with supports in a constructible set

Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
3
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0answers
64 views

Cohomology of boundary of locally symmetric space

Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the ...
2
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0answers
92 views

Cohomology of adelic locally symmetric spaces

I am most probably wrong in asserting as follows. Let $G$ be a connected reductive group over $\mathbb{Q}$, and $S_{K_f} = G(\mathbb{Q}) \backslash G(\mathbb{A}/K_\infty Z(\mathbb{A}) \cdot K_f $ be ...
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175 views

Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?

I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak. Question: Let $M$ be a ...
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0answers
134 views

$\operatorname{Ext}^2(O,\omega)$ as a higher extension on $\mathbb{P}^1 \times \mathbb{P}^1$

Let $X = \mathbb{P}^1 \times \mathbb{P}^1$ over a field $k$ and consider $Ext^2(\mathcal{O}_X,\omega_X)\cong H^2(\omega_X) = H^2(\mathcal{O}_X(-2,-2)) = k$ Let $C = \mathbb{P}^1$. By Kunneth $H^2(\...
6
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0answers
258 views

What is the geometric intuition for the sheaf-theoretic terms “soft”, “fine”, and “flabby”?

The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here). ...
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0answers
311 views

In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?

Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...
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0answers
289 views

Generalities on sheaves - Where can I find the technology that can make this “proof” of Atiyah duality precise?

Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct). Let $X$ ...
4
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0answers
166 views

Dolbeault cohomology of $\text{sl}(2,\mathbb{C})$

Consider the complex Lie group $G=\text{SL}(2,\mathbb{C})$ and let us denote $\Omega$ the sheaf of top holomorphic forms of this group. Are the cohomology spaces $H^{*}(G,\Omega)$ known ? I am ...
3
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0answers
120 views

de Rham isomorphism with holomorphic forms

For a non-compact Riemann surface $X$ there is an isomorphism: $$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$ where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
4
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0answers
206 views

Does hypercohomology of the Koszul complex compute sheaf cohomology?

Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
4
votes
2answers
492 views

Different definition of sheaf cohomology

It could be related to my previous question here. Let $\mathcal F$ be a sheaf on a topological space $X$. Hartshorne in his book on Algebraic geometry defines the sheaf cohomology by $$ H^i(X, \...